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We study a geometric property in Köthe spaces which is called orthogonal uniform convexity (UC┴). It was introduced in [19]. We prove that the class of Köthe spaces with property (UC┴) is a proper subset of the class of uniformly monotone and P-convex Köthe spaces. Next we consider connections between (UC┴) and property (β) of Rolewicz. We shown that the implication (UC┴) → (β) is true in any Köthe sequence space. Moreover, we find criteria for Orlicz function (sequence) spaces to be orthogonally uniformly convex. As a corollary we get that there holds (UC) → (UC┴) → (β) in any Köthe sequence space and the converse of any of these implications is not true. Furthermore, the implications (UC) → (β) → (UC┴) hold in any Köthe function space and the second one cannot be reversed.
Wydawca
Rocznik
Tom
Strony
395--411
Opis fizyczny
Bibliogr. 33 poz.
Twórcy
autor
- Institute of Mathematics, Poznań University of Technology, Piotrowo 3A, 60-965 Poznań, Poland, kolwicz@math.put.poznan.pl
Bibliografia
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- [2] G. Alherk, H. Hudzik, Uniformly non-ln(1) Musielak-Orlicz spaces of Bochner type, Forum Math., 1 (1989) 403-410.
- [3] G. Birkhoff, Lattice Theory, Amer. Math. Soc., Providence, RI 1967.
- [4] S. Chen, Geometry of Orlicz spaces, Dissertationes Math., 356 (1996) 1-204.
- [5] S. Chen, H. Hudzik, On some convexities of Orlicz and Orlicz-Bochner spaces, Comment. Math. Univ. Carolin., 29 (1) (1988) 13-29.
- [6] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936) 396-414.
- [7] Y. Cui, H. Hudzik, T. Zhang, On some geometric properties of certain Köthe sequence spaces, Math. Bohem., (2-3) (1999) 303-314.
- [8] Y. Cui, R. Płuciennik, T. Wang, On property (β) in Orlicz spaces, Arch. Math. (Basel), 69 (1997) 57-69.
- [9] K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge 1990.
- [10] H. Hudzik, Strict convexity of Musielak-Orlicz spaces with Luxemburg norm, Bull. Ac. Pol.: Math., 39 (5-6) (1981) 235-246.
- [11] H. Hudzik, Uniform convexity of Musielak-Orlicz spaces with Luxemburg’s norm, Comm. Math., 23 (1983) 23-30.
- [12] H. Hudzik, Uniformly non-ln(1) Orlicz spaces with Luxemburg norm, Studia Math., 81 (3) (1985) 271-284.
- [13] H. Hudzik, A. Kamińska, M. Mastyło, Monotonicity and rotundity properties in Banach lattices, Rocky Mountain J. Math., 30 (3) (2000) 933-950.
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- [16] A. Kamińska, Rotundity of Orlicz-Musielak sequence spaces, Bull. Acad. Polon. Sci. Math., 29 (3-4) (1981) 137-144.
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- [19] P. Kolwicz, On property (β) in Banach lattices, Calderón-Lozanowskiĭ and Orlicz-Lorentz spaces, Proc. Indian Acad. Sci. (Math. Sci.), 111 (3) (2001) 319-336.
- [20] P. Kolwicz, On property (β) in Köthe-Bochner function spaces, Czechoslovak Math. J., submitted.
- [21] P. Kolwicz, H. Hudzik, A note on P-convexity of Orlicz and Musielak-Orlicz spaces of Bochner type, in: Proc. Fifth Conf, of Function Spaces (Poznań 1998), Lecture Notes in Math., 213, Marcel Decker Inc., New York (2000) 181-192.
- [22] P. Kolwicz, H. Hudzik, On property (β) of Rolewicz in Köthe-Bochner sequence spaces, Studia Math., submitted.
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Bibliografia
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bwmeta1.element.baztech-article-BAT2-0001-1280